Adaptive Numerical Solution of PDEs (eBook)

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Peter Deuflhard and Martin Weiser, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany.

Preface 5
Outline 13
1 Elementary Partial Differential Equations 17
1.1 Laplace and Poisson Equation 17
1.1.1 Boundary Value Problems 18
1.1.2 Initial Value Problem 22
1.1.3 Eigenvalue Problem 24
1.2 Diffusion Equation 27
1.3 Wave Equation 30
1.4 Schrödinger Equation 35
1.5 Helmholtz Equation 38
1.5.1 Boundary Value Problems 38
1.5.2 Time-harmonic Differential Equations 39
1.6 Classification 41
1.7 Exercises 43
2 Partial Differential Equations in Science and Technology 46
2.1 Electrodynamics 46
2.1.1 Maxwell Equations 46
2.1.2 Optical Model Hierarchy 49
2.2 Fluid Dynamics 52
2.2.1 Euler Equations 53
2.2.2 Navier-Stokes Equations 56
2.2.3 Prandtl’s Boundary Layer 61
2.2.4 Porous Media Equation 63
2.3 Elastomechanics 64
2.3.1 Basic Concepts of Nonlinear Elastomechanics 64
2.3.2 Linear Elastomechanics 68
2.4 Exercises 71
3 Finite Difference Methods for Poisson Problems 74
3.1 Discretization of Standard Problem 74
3.1.1 Discrete Boundary Value Problems 75
3.1.2 Discrete Eigenvalue Problem 80
3.2 Approximation Theory on Uniform Grids 83
3.2.1 Discretization Error in L2 85
3.2.2 Discretization Error in L? 88
3.3 Discretization on Nonuniform Grids 90
3.3.1 One-dimensional Special Case 90
3.3.2 Curved Boundaries 92
3.4 Exercises 95
4 Galerkin Methods 98
4.1 General Scheme 98
4.1.1 Weak Solutions 98
4.1.2 Ritz Minimization for Boundary Value Problems 101
4.1.3 Rayleigh-Ritz Minimization for Eigenvalue Problems 105
4.2 Spectral Methods 107
4.2.1 Realization by Orthogonal Systems 108
4.2.2 Approximation Theory 112
4.2.3 Adaptive Spectral Methods 115
4.3 Finite Element Methods 120
4.3.1 Meshes and Finite Element Spaces 120
4.3.2 Elementary Finite Elements 123
4.3.3 Realization of Finite Elements 133
4.4 Approximation Theory for Finite Elements 140
4.4.1 Boundary Value Problems 140
4.4.2 Eigenvalue Problems 143
4.4.3 Angle Condition for Nonuniform Meshes 148
4.5 Exercises 151
5 Numerical Solution of Linear Elliptic Grid Equations 155
5.1 Direct Elimination Methods 156
5.1.1 Symbolic Factorization 157
5.1.2 Frontal Solvers 159
5.2 Matrix Decomposition Methods 162
5.2.1 Jacobi Method 164
5.2.2 Gauss-Seidel Method 166
5.3 Conjugate Gradient Method 168
5.3.1 CG-Method as Galerkin Method 168
5.3.2 Preconditioning 171
5.3.3 Adaptive PCG-method 175
5.3.4 A CG-variant for Eigenvalue Problems 177
5.4 Smoothing Property of Iterative Solvers 182
5.4.1 Illustration for the Poisson Model Problem 182
5.4.2 Spectral Analysis for Jacobi Method 186
5.4.3 Smoothing Theorems 187
5.5 Iterative Hierarchical Solvers 192
5.5.1 Classical Multigrid Methods 194
5.5.2 Hierarchical-basis Method 202
5.5.3 Comparison with Direct Hierarchical Solvers 205
5.6 Power Optimization of a Darrieus Wind Generator 206
5.7 Exercises 212
6 Construction of Adaptive Hierarchical Meshes 215
6.1 A Posteriori Error Estimators 215
6.1.1 Residual Based Error Estimator 218
6.1.2 Triangle Oriented Error Estimators 223
6.1.3 Gradient Recovery 227
6.1.4 Hierarchical Error Estimators 231
6.1.5 Goal-oriented Error Estimation 234
6.2 Adaptive Mesh Refinement 235
6.2.1 Equilibration of Local Discretization Errors 236
6.2.2 Refinement Strategies 241
6.2.3 Choice of Solvers on Adaptive Hierarchical Meshes 245
6.3 Convergence on Adaptive Meshes 245
6.3.1 A Convergence Proof 246
6.3.2 An Example with a Reentrant Corner 248
6.4 Design of a Plasmon-Polariton Waveguide 252
6.5 Exercises 256
7 Adaptive Multigrid Methods for Linear Elliptic Problems 258
7.1 Subspace Correction Methods 258
7.1.1 Basic Principle 259
7.1.2 Sequential Subspace Correction Methods 262
7.1.3 Parallel Subspace Correction Methods 267
7.1.4 Overlapping Domain Decomposition Methods 271
7.1.5 Higher-order Finite Elements 278
7.2 Hierarchical Space Decompositions 283
7.2.1 Decomposition into Hierarchical Bases 284
7.2.2 L2-orthogonal Decomposition: BPX 290
7.3 Multigrid Methods Revisited 294
7.3.1 Additive Multigrid Methods 294
7.3.2 Multiplicative Multigrid Methods 298
7.4 Cascadic Multigrid Methods 301
7.4.1 Theoretical Derivation 301
7.4.2 Adaptive Realization 307
7.5 Eigenvalue Problem Solvers 312
7.5.1 Linear Multigrid Method 313
7.5.2 Rayleigh Quotient Multigrid Method 315
7.6 Exercises 318
8 Adaptive Solution of Nonlinear Elliptic Problems 322
8.1 Discrete Newton Methods for Nonlinear Grid Equations 323
8.1.1 Exact Newton Methods 324
8.1.2 Inexact Newton-PCG Methods 328
8.2 Inexact Newton-Multigrid Methods 331
8.2.1 Hierarchical Grid Equations 331
8.2.2 Realization of Adaptive Algorithm 333
8.2.3 An Elliptic Problem Without a Solution 337
8.3 Operation Planning in Facial Surgery 340
8.4 Exercises 343
9 Adaptive Integration of Parabolic Problems 345
9.1 Time Discretization of Stiff Differential Equations 345
9.1.1 Linear Stability Theory 346
9.1.2 Linearly Implicit One-step Methods 352
9.1.3 Order Reduction 359
9.2 Space-time Discretization of Parabolic PDEs 365
9.2.1 Adaptive Method of Lines 366
9.2.2 Adaptive Method of Time Layers 374
9.2.3 Goal-oriented Error Estimation 383
9.3 Electrical Excitation of the Heart Muscle 386
9.3.1 Mathematical Models 386
9.3.2 Numerical Simulation 387
9.4 Exercises 390
A Appendix 392
A.1 Fourier Analysis and Fourier Transform 392
A.2 Differential Operators in ?3 393
A.3 Integral Theorems 395
A.4 Delta-Distribution and Green's Functions 399
A.5 Sobolev Spaces 404
A.6 Optimality Conditions 409
B Software 410
B.1 Adaptive Finite Element Codes 410
B.2 Direct Solvers 411
B.3 Nonlinear Solvers 411
Bibliography 413
Index 427